Logical differential calculus for calculation of Birnbaum importance of non-coherent system

Abstract

The Birnbaum importance is one of the measures often used in importance analysis. Many algorithms for the calculation of this measure have been proposed for coherent systems. Several interpretations of the Birnbaum importance for non-coherent systems also exist. These interpretations are usually based on the prime implicants of the structure function of the system. However, identification of the prime implicants is a complex problem. In this paper, we develop a method for computation of Birnbaum importance using Boolean expression (algebraic representation) and truth table (matrix representation). Both of these representations are calculated based on Logical Differential Calculus, in particular, Boolean derivatives. The Boolean expression derived based on Boolean derivatives is equal to known definitions of the Birnbaum importance. The matrix approach for the calculation of the Birnbaum importance based on Boolean derivatives is novel. This approach can be used for calculation of the Birnbaum importance for the structure function represented by truth table and does not need transformation of the structure function into Boolean expression. An important advantage of the proposed approach (in algebraic or matrix form) is a possibility to define the Birnbaum importance for the evaluation of the non-coherent system failure (restore) with respect to simultaneous changes of states of several system components.